Introduction to Vectors
This lesson will serve as an introduction to vectors and how they operate. Of course, this raises the obvious question: What are vectors?
A vector is any quantity that has both direction and magnitude. You'll likely just see them as an arrow drawn pointing in a certain direction and with a certain length (its magnitude).
Firstly, two vectors are considered equal to each other if and only if they have the same direction (usually given by the angle it makes with the horizontal) and its magnitude (its length). This means that vectors do not have to start at the same point to be considered equal. This concept is illustrated in the animation below:
Now that we understand when vectors are equal, we can begin to modify them. The first thing we can do is multiply them by constants (known as scalars in this context). The following things can happen to the vector:
If the absolute value of the scalar is greater than 1, the vector's magnitude will grow.
If the absolute value of the scalar is less than 1, the vector's magnitude will shrink.
If the absolute value of the scalar is 1, the vector's magnitude will not change.
If the scalar is less than 0, the vector will flip directions.
If the scalar is 0, the vector will essentially become a point.
We can also add two vectors together. To add one vector to another, it's as simple as connecting the tip of the first to the tail of the second, then drawing the vector that runs from the tail of the first to the tip of the second.
To subtract one vector from another, flip the direction of the second vector and follow the same process.
Now, we will introduce the concept of unit vectors. For our purposes, there are two unit vectors, always called i and j. i has a magnitude of 1 and is horizontal, while j has a magnitude of 1 and is vertical.
We are introducing these vectors because we can use them to describe any vector by simply multiplying each of them by some scalar and adding them together.
In general, any vector can be written as ai + bj for any scalars a and b, meaning any vector can be characterized by its horizontal and vertical components alone.
Of course, at the beginning of this lesson, we mentioned that vectors can be characterized by their direction (angle made with the horizontal) and magnitude. Thanks to unit vectors, we can now do this in a meaningful way.
In words, we can recognize that the vector forms a right triangle, with its component vectors acting as the legs and the vector itself acting as the hypotenuse. Since the sine of the angle theta is opposite over hypotenuse (SOH-CAH-TOA), it is equal to the magnitude of the vector's vertical component over the magnitude of the vector itself. Similarly, the cosine is the magnitude of the vector's horizontal component over the magnitude of the vector itself. Multiplying by the magnitude of the vector on both sides, we can find two expressions with the magnitude of the vector and the angle it makes with the horizontal that we can plug into ai + bj.
Finally, there is the matter of the dot product. Vectors aren't quite like the scalars we're used to. Adding and subtracting them and multiplying them by scalars has worked so far, but you may have already started wondering what it may mean to multiply two vectors together. You may have tried to define vector multiplication taking (ai + bj)(ci + dj) and expanding it, but you end up multiplying i and j together, which only leaves you with the question you tried to answer in the first place. There are a few definitions of "vector multiplication", but one of them is the cross product.
The dot product of a vector ai + bj and ci + dj is given by ac + bd. In other words, you multiply the two horizontal components, multiply the two vertical components, and add the two to find the dot product. As you may have noticed, this is a scalar, not a vector. Another interesting fact about the dot product is that it is also equal to the magnitude of the first vector times the magnitude of the second vector times the cosine of the angle between them. Mathematically,
Equating the two, this gives us an easy way to find the angle between two vectors just using their horizontal and vertical components:
The following animation is a visual proof of this convenient fact by treating each vector as the sides of a triangle and using the law of cosines near the end. If you've forgotten or haven’t yet learned the law of cosines, it states that for a triangle with known sides a, b and known angle C opposite to side c, the following is true:
